The square numbers are the numbers 1· 1,  2· 2,  3· 3, and so on.  The following are the first ten square numbers and their roots.

The square root of 1 is 1.  The square root of 4 is 2.   The square root of 9 is 3.  And so on.

We write

= 5

"The square root of 25 is 5."

This mark is called the radical sign (after the Latin radix = root).  The number under the line is called the radicand.  In the example, 25 is the radicand.

Problem 1.   Evaluate the following.

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a)     8       b)     13       c)     20

The fundamental algebraic property of the square root radical is the following:

A radical multiplied by itself

produces the radicand.

Problem 2.   Evaluate the following.

a)   ·   = 3            b)   ()²  = 5            c)   ()²  = a + b

Example 1.   Solve this equation:

Example 2.   Solve this equation:

	x²	=	10.
Solution.	x	=	or  −.

Because ()² = 10,  and  (−)² = 10.

In other words, if an equation looks like this:

Solution.	x²	=	a,
then its solution will look like this:
Solution.	x	=	or  −.

We often use the double sign  ± ("plus or minus")  and write:

Solution.

x

=

±.

Problem 3.   Solve for x.

a)   x² = 9   x = ±3       b)   x² = 8   x = ±       c)   x² = q   x = ±

Rational and Irrational Numbers

A rational number is any whole number, any fraction, any mixed number, or any decimal;  positive, negative, or 0.  The rational numbers are the ordinary knowable numbers of arithmetic.

That is what a rational number is.  As for what it looks

integers (b ≠ 0).

Problem 4.   Which of the following numbers are rational?

All of them!

An example of a number that is not rational is .   is not any whole number, it is not any fraction -- there is no fraction which, when multiplied by itself, will produce 2 -- and it is not any decimal.  Now,

1.414

(The wavy equal sign means "is approximately".)  How could we know that?  By multiplying 1.414 by itself.  If we do, we get 1.999396 -- which is almost 2.  It should be clear that no decimal multiplied by itself can ever be exactly 2.00000000000000.  Because if the decimal ends in 1, then its square will end in 1.  If the decimal ends it 2, its square will end in 4.  And so on.  No decimal -- no rational number -- multiplied by itself can ever produce 2.  Therefore, is called an irrational number.

Which square roots are rational and which are irrational?

Only the square roots of square numbers are rational.  All the rest are irrational.

= 1  Rational

 Irrational

 Irrational

= 2  Rational

,  ,  ,  Irrational

= 3  Rational

And so on.

Rationalize a denominator

Multiply radicals

Multiply radicals as follows:

Problem 6.    a)   ·  =             b)   ·  = = 4